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title: W14 NotesPolar coordinates are pairs of numbers
Polar coordinates have many redundancies: unlike Cartesian which are unique!
Polar coordinates cannot be added: they are not vector components!
⚠️ The transition formulas
Equations (as well as points) can also be converted to polar.
For
For
Compute the polar coordinates of
Solution
For
Next compute:
This angle is in Quadrant IV. We add
The radius is of course
For
Next compute:
This is the correct angle because Quadrant IV is SAFE. So the point in polar is
For example, let’s convert a shifted circle to polar. Say we have the Cartesian equation:
Then to find the polar we substitute
So this shifted circle is the polar graph of the polar function
To draw the polar graph of some function, it can help to first draw the Cartesian graph of the function. (In other words, set
This Cartesian graph may be called a graphing tool for the polar graph.
A limaçon is the polar graph of
Any limaçon shape can be obtained by adjusting
Limaçon satisfying
Limaçon satisfying
Limaçon satisfying
Limaçon satisfying
Limaçon satisfying
Limaçon satisfying
Transitions between limaçon types,
Notice the transition points at
The flat spot occurs when
The cusp occurs when